mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'

Tactic:
have A : ContinuousWithinAt (fun r => f r - f a) (Ioi a) a := by
      refine ContinuousWithinAt.sub ?_ continuousWithinAt_const
      exact (f.right_continuous a).mono Ioi_subset_Ici_self

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a'
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
hl : f.length (s i) < f.length (s i) + ↑(ε' i)
| f.length (s i) < f.length (s i) + ↑(ε' i)

Tactic:
lhs
      rw [length_eq]

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
hl : f.length (s i) < f.length (s i) + ↑(ε' i)
| ⨅ a, ⨅ b, ⨅ (_ : s i \ botSet ⊆ Ioc a b), ofReal (↑f b - ↑f a)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ (inducedOuterMeasure (fun U hU ↦ μ.innerContent { carrier := U, is_open' := hU }) ⋯ ⋯) ↑K

Tactic:
inducedOuterMeasure_eq_iInf

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ μ K ≤ ⨅ t, ⨅ (ht : IsOpen t), ⨅ (_ : ↑K ⊆ t), μ.innerContent { carrier := t, is_open' := ht }

case PU
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ ∀ ⦃f : ℕ → Set G⦄, (∀ (i : ℕ), IsOpen (f i)) → IsOpen (⋃ i, f i)

case msU
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ ∀ ⦃f : ℕ → Set G⦄ (hm : ∀ (i : ℕ), IsOpen (f i)),
    μ.innerContent { carrier := ⋃ i, f i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := f i, is_open' := ⋯ }

case m_mono
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
K : Compacts G
⊢ ∀ ⦃s₁ s₂ : Set G⦄ (hs₁ : IsOpen s₁) (hs₂ : IsOpen s₂),
    s₁ ⊆ s₂ → μ.innerContent { carrier := s₁, is_open' := hs₁ } ≤ μ.innerContent { carrier := s₂, is_open' := hs₂ }
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
A : Set G
hA : μ.outerMeasure A ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε

Tactic:
rcases inducedOuterMeasure_exists_set _ μ.innerContent_iUnion_nat μ.innerContent_mono hA
      (ENNReal.coe_ne_zero.2 hε) with
    ⟨U, hU, h2U, h3U⟩
  exact ⟨⟨U, hU⟩, h2U, h3U⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ∑' (i : ℕ), f.length (s i) + ↑δ + ↑δ = ∑' (i : ℕ), f.length (s i) + ↑ε

Tactic:
by simp [δ, add_assoc, ENNReal.add_halves]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case h.w
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g₀ ∈ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}

Tactic:
simp only [Finset.mem_filter, h1g₀, true_and]; use g
      simp [hg, h2g₀]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
x : R
this✝ : Nonempty R
val✝ : OrderBot R
hf : Tendsto (↑f) (pure ⊥) (𝓝 l)
this : Iic x = Icc ⊥ x
⊢ f.measure (Icc ⊥ x) = ofReal (↑f x - l)

Tactic:
measure_Icc,

Goal state after:
case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
x : R
this✝ : Nonempty R
val✝ : OrderBot R
hf : Tendsto (↑f) (pure ⊥) (𝓝 l)
this : Iic x = Icc ⊥ x
⊢ ofReal (↑f x - leftLim ↑f ⊥) = ofReal (↑f x - l)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
x : R
this✝ : Nonempty R
val✝ : OrderTop R
hf : Tendsto (↑f) (pure ⊤) (𝓝 l)
this : Ici x = Icc x ⊤
⊢ f.measure (Icc x ⊤) = ofReal (l - leftLim (↑f) x)

Tactic:
measure_Icc,

Goal state after:
case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
l : ℝ
x : R
this✝ : Nonempty R
val✝ : OrderTop R
hf : Tendsto (↑f) (pure ⊤) (𝓝 l)
this : Ici x = Icc x ⊤
⊢ ofReal (↑f ⊤ - leftLim (↑f) x) = ofReal (l - leftLim (↑f) x)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hs
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
⊢ ∀ x ∈ s, 1 ≤ ENNReal.ofReal (f x)

Tactic:
intro x

Goal state after:
case hs
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
f_int : Integrable f μ
f_nonneg : 0 ≤ᵐ[μ] f
s : Set X
hs : ∀ x ∈ s, 1 ≤ f x
x : X
⊢ x ∈ s → 1 ≤ ENNReal.ofReal (f x)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x : F
⊢ ↑(normSq x) = ⟪x, x⟫

Tactic:
by
  rw [ext_iff]
  exact ⟨by simp only [ofReal_re, normSq], by simp only [inner_self_im, ofReal_im]⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ (a : X), ENNReal.ofReal (g a) ≤ 1

Tactic:
intro x

Goal state after:
case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ ENNReal.ofReal (g x) ≤ 1
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0

Tactic:
intro x

Goal state after:
case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
⊢ x ∈ sᶜ → ENNReal.ofReal (g x) = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
⊢ ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖

Tactic:
have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_mem_range.2 fun _ => norm_nonneg _⟩

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
⊢ ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫ = ‖x‖ ^ 2

Tactic:
by
  rw [pow_two, inner_self_eq_norm_mul_norm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) = ‖x + y‖ ↔ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
sqrt_eq_iff_mul_self_eq,

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ = ‖x + y‖ * ‖x + y‖ ↔ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

case hx
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ 0 ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

case hy
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ 0 ≤ ‖x + y‖
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) = ‖x - y‖ ↔ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

Tactic:
sqrt_eq_iff_mul_self_eq,

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ = ‖x - y‖ * ‖x - y‖ ↔ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

case hx
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ 0 ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖

case hy
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ 0 ≤ ‖x - y‖
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case inl.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = v i
hj : w j = v j
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0

Tactic:
replace hv := hv i j

Goal state after:
case inl.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = v i
hj : w j = v j
hv : ⟪v i, v j⟫ = if i = j then 1 else 0
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case inl.inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = v i
hj : w j = -v j
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0

Tactic:
replace hv := hv i j

Goal state after:
case inl.inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = v i
hj : w j = -v j
hv : ⟪v i, v j⟫ = if i = j then 1 else 0
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = -v i
hj : w j = v j
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0

Tactic:
replace hv := hv i j

Goal state after:
case inr.inl
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = -v i
hj : w j = v j
hv : ⟪v i, v j⟫ = if i = j then 1 else 0
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case inr.inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = -v i
hj : w j = -v j
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0

Tactic:
replace hv := hv i j

Goal state after:
case inr.inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v w : ι → E
hw : ∀ (i : ι), w i = v i ∨ w i = -v i
i j : ι
hi : w i = -v i
hj : w j = -v j
hv : ⟪v i, v j⟫ = if i = j then 1 else 0
⊢ ⟪w i, w j⟫ = if i = j then 1 else 0
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
⊢ (innerSL ℝ).flip = innerSL ℝ

Tactic:
ext v w
  exact real_inner_comm _ _

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ∃ y ∈ ⋂ n, f n, y ∈ closedEBall x ε

Tactic:
let F : ℕ → X × ℝ≥0∞ := fun n =>
    Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2)

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
⊢ ∃ y ∈ ⋂ n, f n, y ∈ closedEBall x ε
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
n : ℕ
⊢ 0 < r n

Tactic:
induction n with
    | zero => exact lt_min εpos (Bpos 0)
    | succ n hn => exact Hpos n (c n) (r n) hn.ne'

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫ = ∑ k ∈ s, ⟪v k, x⟫ * ⟪x, v k⟫
h₃ : ∀ (z : 𝕜), re (z * (starRingEnd 𝕜) z) = ‖z‖ ^ 2
⊢ ∑ i ∈ s, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2

Tactic:
suffices hbf : ‖x - ∑ i ∈ s, ⟪v i, x⟫ • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i ∈ s, ‖⟪v i, x⟫‖ ^ 2 by
    rw [← sub_nonneg, ← hbf]
    simp only [norm_nonneg, pow_nonneg]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
x : E
v : ι → E
s : Finset ι
hv : Orthonormal 𝕜 v
h₂ : ∑ i ∈ s, ∑ j ∈ s, ⟪v i, x⟫ * ⟪x, v j⟫ * ⟪v j, v i⟫ = ∑ k ∈ s, ⟪v k, x⟫ * ⟪x, v k⟫
h₃ : ∀ (z : 𝕜), re (z * (starRingEnd 𝕜) z) = ‖z‖ ^ 2
⊢ ‖x - ∑ i ∈ s, ⟪v i, x⟫ • v i‖ ^ 2 = ‖x‖ ^ 2 - ∑ i ∈ s, ‖⟪v i, x⟫‖ ^ 2
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ ‖reCLM‖ = 1

Tactic:
apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)

Goal state after:
K : Type u_1
inst✝ : RCLike K
⊢ 1 ≤ ‖reLm.mkContinuous 1 ⋯‖
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ K ⊓ Kᗮ ≤ ⊥

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
x : E
⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
⊢ (T + S).IsSymmetric

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(T + S) x, y⟫ = ⟪x, (T + S) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
⊢ (T - S).IsSymmetric

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(T - S) x, y⟫ = ⟪x, (T - S) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
c : 𝕜
hc : (starRingEnd 𝕜) c = c
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
⊢ (c • T).IsSymmetric

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
c : 𝕜
hc : (starRingEnd 𝕜) c = c
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x : E
⊢ ∀ (y : E), ⟪(c • T) x, y⟫ = ⟪x, (c • T) y⟫
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : ClosedSubmodule 𝕜 E
⊢ K ⊓ Kᗮ ≤ ⊥

Tactic:
intro x

Goal state after:
𝕜 : Type u_4
E : Type u_5
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : ClosedSubmodule 𝕜 E
x : E
⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ ∀ (x : E), ‖K.reflectionLinearEquiv x‖ = ‖x‖

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
⊢ ‖K.reflectionLinearEquiv x‖ = ‖x‖
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : RCLike 𝕜
E : Type u_4
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace 𝕜 E'
f : E ≃ₗᵢ[𝕜] E'
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E'
⊢ (map (↑f.toLinearEquiv) K).reflection x = f (K.reflection (f.symm x))

Tactic:
by
  simp [reflection_apply, starProjection_map_apply f K x]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: case intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝² : Preorder ι
U : ι → Submodule 𝕜 E
inst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection
inst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection
hU : Monotone U
x : E
h : atTop.NeBot
left✝ : Nonempty ι
right✝ : IsDirectedOrder ι
y : E := (⨆ i, U i).topologicalClosure.starProjection x
proj_x : ∀ (i : ι), (U i).orthogonalProjectionOnto x = (U i).orthogonalProjectionOnto y
⊢ ∀ ε > 0, ∃ I, ∀ i ≥ I, ‖(U i).starProjection y - y‖ < ε

Tactic:
intro ε hε

Goal state after:
case intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝² : Preorder ι
U : ι → Submodule 𝕜 E
inst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection
inst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection
hU : Monotone U
x : E
h : atTop.NeBot
left✝ : Nonempty ι
right✝ : IsDirectedOrder ι
y : E := (⨆ i, U i).topologicalClosure.starProjection x
proj_x : ∀ (i : ι), (U i).orthogonalProjectionOnto x = (U i).orthogonalProjectionOnto y
ε : ℝ
hε : ε > 0
⊢ ∃ I, ∀ i ≥ I, ‖(U i).starProjection y - y‖ < ε